Boris Johnson should resign after the Supreme Court found his suspension of Parliament is unlawful, First Minister Mark Drakeford has said.
The court's justices found that, in effect, the five-week prorogation never took place.
It is a "victory for the rule of law", Mr Drakeford said.
The prime minister said he would "respect the verdict", but that it had made negotiating with the EU more difficult.
Brexiteers called for a general election, with Conservative Monmouth MP David Davies saying the prime minister was facing a "pro-EU establishment".
House of Commons Speaker John Bercow has said Parliament will resume at 11:30 BST on Wednesday.
Mr Drakeford said: "The prime minister tried to play fast and loose with our constitution. The unanimous decision by the Supreme Court is a huge victory for the rule of law.
"Parliament was never prorogued and must now hold this government to account."
Answering an urgent question in the Senedd, he told AMs: "The Prime Minister has been found to have acted unlawfully and in a way deliberately designed to subvert our democracy.
"How does anybody in that position think that they can carry on?"
Prime Minister Boris Johnson controversially announced on 28 August that Parliament would be suspended until mid-October, causing a row with opposition politicians.
The Welsh Government argued it got in the way of the work of the assembly - opposition MPs complained it meant they were unable to properly scrutinise the UK government ahead of Brexit.
Ruling unanimously, the Supreme Court's 11 judges found the PM's advice to the Queen was "unlawful, void and of no effect" when he asked the Queen to suspend Parliament.
The decision "had the effect of frustrating or preventing" parliament from carrying out its functions, it said.
Speaking in New York, where he is attending the United Nations General Assembly, Mr Johnson called the ruling an "unusual judgement".
"I don't think this was the right decision," he told reporters.
"The prerogative of prorogation has been used for centuries without this kind of challenge.
"There are a lot of people who basically want to stop this country from coming out of the EU and we have a Parliament that is unable to be prorogued and doesn't want to have an election.
"I think it is time we took things forward."
The prime minister said getting a deal was "not made much easier with these sort of things in Parliament or the courts", but insisted the UK would still leave the EU on 31 October.
A No 10 source said the Supreme Court had "made a serious mistake in extending its reach to these political matters", and had "made it clear that its reasons [were] connected to the Parliamentary disputes over, and timetable for" Brexit.
Independent MP and former Conservative defence minister Guto Bebb said Justice Secretary Robert Buckland and Attorney General Geoffrey Cox "should both resign immediately" if the source's comments reflected the official government position.
Asked to respond to the comments on BBC Politics Live, Mr Bebb said: "If they are willing to be part of an administration which is attacking the integrity of 11 Supreme Court judges, then I do think that the position of the Justice Secretary and the Attorney General is now untenable.
"I think that the responsibility that they have to protect the judiciary is clearly not compatible with serving in a government that decides to attack the legal system in a manner which you have just articulated."
Any normal Prime Minister would – as a matter of honour – tender their resignation after such a unanimous verdict from the UK’s highest court. It is the final straw in a pitiful episode for the country.— Mark Drakeford (@fmwales) September 24, 2019
The Supreme Court made its judgement following cases in the English and Scottish courts, the former backed by the Welsh Government, the latter supported by a number of Welsh MPs.
Welsh Conservative assembly group leader Paul Davies said: "The Conservative Party respects the rule of law and the outcome of the Supreme Court proceedings. However, we must remember that the Welsh and British people voted to leave the European Union and we must respect their wishes.
He called for an election "as soon as possible". In the Senedd he claimed Brexit impasse had been created by Labour MPs frustrating the Brexit process "against the will of the Welsh and British people".
Labour's MP for Cardiff South and Penarth Stephen Doughty told the BBC: "He's lied to the Queen, he's acted unlawfully and he's played fast and loose with our constitution and our basic principles.
"The Supreme Court unanimously is upholding our constitution. What he's done is he prevented us as members of Parliament from not only doing our job on Brexit but on so many other issues, from pensions to climate change to health to education."
Plaid Cymru MPs also backed the Scottish case. Party leader Adam Price said: "There is only one response now acceptable by the prime minister to this damning and unanimous judgement by the Supreme Court: Resign."
"After getting rid of this dishonest prime minister Parliament must now move to end the chaos and uncertainty and take it back to the people in a People's Vote so we can finally move on from this dark chapter in our politics."
Westminster Plaid leader Liz Saville Roberts said: "The prime minister has shown himself to be no better than a tin-pot dictator, shutting down democracy to avoid scrutiny."
Some MPs, including Swansea East MP Carolyn Harris, tweeted pictures of themselves returning to the green benches of the Commons.
Welsh Liberal Democrat leader Jane Dodds welcomed the "damning judgement from the courts that confirms the prorogation of Parliament was unlawful and politically motivated".
"Parliament must be recalled now so we can get on with our jobs," she said. "With the Brexit clock still ticking down there isn't a moment to lose."
Monmouth Conservative MP David Davies defended the Tory leader, saying he stood with Mr Johnson "who is doing everything possible to deliver on the clear result of a referendum in the face of a powerful pro EU establishment".
"They want the PM to resign - but they don't want an election," he said.
He added: "The PM has given opposition parties two chances for an election.... we should offer a third."
Analysis by Felicity Evans, BBC Wales political editor
The big question now is: what do opposition MPs do when Parliament returns? Lots of inter-party and cross-party meetings are taking place to decide on a strategy.
Having called for Boris Johnson to resign - do opposition parties table a vote of no confidence?
I'm hearing that despite some initial enthusiasm in some quarters, Plaid, Labour and the Lib Dems seem to be agreeing that it's too risky. They think it's a "trap" because it could lead to a no-deal departure from the EU at the end of the month if an election is triggered.
I'm being briefed there may be censure motions or more SO24 motions - remember them? They allowed MPs take control of Parliament a few weeks ago. But I'm also hearing that the real fireworks may be reserved for next week when more MPs have been able to return from their travels.
And what will the government do? From his comments in New York, it seems pretty clear that the PM will not be resigning and that he could try to suspend Parliament again, presumably for a shorter time.
The Conservative AM David Melding tweeted earlier that the Supreme Court judgement means: "We are in new territory." True enough, and no one has a map.
On Monday, Mr Johnson refused to rule out suspending Parliament for a second time if the Supreme Court ruled against him.
Former First Minister Carwyn Jones said that would put the Queen in an "invidious position".
Following the ruling, he told BBC Wales: "If you were the Palace, you would be saying: Having been told that the advice was improper, how can we sure that the advice is proper this time round?
That would put the Queen in an invidious position of having to agree to something potentially on the basis of advice that's actually in effect illegal."